The prefix palindromic length $p_{\mathbf{u}}(n)$ of an infinite word $\mathbf{u}$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $\mathbf{u}$. This function is surprisingly difficult to study; in particular, the conjecture that $p_{\mathbf{u}}(n)$ can be bounded only if $\mathbf{u}$ is ultimately periodic is open since 2013. A more recent conjecture concerns the prefix palindromic length of the period doubling word: it seems that it is not $2$-regular, and if it is true, this would give a rare if not unique example of a non-regular function of a $2$-automatic word. For some other $k$-automatic words, however, the prefix palindromic length is known to be $k$-regular. Here we add to the list of those words the Sierpinski word $\mathbf{s}$ and give a complete description of $p_{\mathbf{s}}(n)$.

翻译：字前的硬度长度 $p ⁇ mathbf{u{{{{{{{{}}}(n)$(n)$这个无限的单词$\ mathbf{u}$\\ mathbf{u}$(maxbf}}$(m)$(n)$) 是表达长度$$\ mathbbf{u}$美元前缀的最小数量。 这个函数令人惊讶地难以研究; 特别是, $p ⁇ mathbf{u}(n)$(n) 的猜想, 只有当$\ mathbf{u} 最终是定期才开始开放时, 才会被约束; 较近的猜想涉及该周期翻倍单词的前缀等长 : 它似乎不是普通的 $2$, 如果是 $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\